Many classes of devices function based on the structure, topology, and other properties of grain boundaries or interfaces. Examples include varistors, PTCR thermistors, diodes, chemical sensors, and solar cells. The properties of interfaces have been extensively studied by macroscopic techniques, such as dc transport measurements and impedance spectroscopy, etc. These techniques address averaged properties of interfaces and little or no information is obtained about the properties of an individual interface. Recently, a number of approaches have been suggested to isolate individual grain boundaries using micropatterned contacts or bicrystal samples. A number of works accessing current-voltage (I-V) characteristic of single interfaces also have been reported; however, a major limitation of such techniques is a preset contact pattern, which does not yield spatially resolved information. Moreover, contact resistance and contact capacitance are included in the measurements, which may decrease accuracy and complicate data interpretation. Scanning probe microscopy (SPM) techniques have been successfully used to detect stray fields over Schottky double barriers and to image potential drops at laterally biased grain boundaries; however, the information provided by SPM has been limited to static or dc transport properties of grain boundaries.
SPM techniques based on the detection of tip-surface capacitance and dc resistivity are well known, and some are capable of detecting the frequency dependence of tip-surface impedance. However, such techniques do not quantify the local impedance of an interface normal to the surface, i.e., local characterization of ac transport properties of an interface.
Further, some techniques provide force gradient images of interfaces, such as conventional magnetic-force microscopy (MFM). MFM is a dual pass technique based on detecting the dynamic response of a mechanically driven cantilever a magnetic field. During a first pass, the ferromagnetic tip of the cantilever acquires a surface topology profile of a sample in an intermittent contact mode. Then, during a second pass, the cantilever is driven mechanically and the surface topographic profile is retraced at a predefined tip-to-sample surface separation. The magnetic force Fmagn between the tip and the sample surface varies along the length of the sample, thereby causing a change in cantilever resonant frequency that is proportional to the force gradient and is given by:
                                                       Δ                    ⁢                                          ⁢          ω                =                                            ω              o                                      2              ⁢              k                                ⁢                                    ⅆ                                                F                  magn                                ⁡                                  (                  z                  )                                                                    ⅆ              z                                                          Equation        ⁢                                  ⁢        1            where k is the cantilever spring constant, ωo is the resonant frequency of the cantilever, and z is the tip to surface separation distance. Resonant frequency shift, Δω, data is collected and arranged as a MFM image of the sample. Image quantification in terms of surface and tip properties is complex due to the non-local character of the tip-surface interactions. In one point probe approximation, the magnetic state of the tip is described by its effective first and second order multipole moments. The force acting on the probe is given by:F=uo(q+m∇)H  Equation 2where q and m are effective probe monopole and dipole moments, respectively, H is the stray field above the surface of the sample, and uo is the magnetic permeability of a vacuum (1.256×10−6H/m). The effective monopole and dipole moments of the tip can be obtained by calibrating against a standard system, for example, micro-fabricated coils or lines carrying a known current. These can be used to quantify the MFM data. However, the total force gradient over the sample may be affected by electrostatic interactions.
Therefore, a system and method that quantifies the local impedance of the interface normal to the sample surface and that overcomes errors introduced by electrostatic interactions would be desirable.